Engineering Optimization: Theory and Practice, Fourth Edition

2.2 Single-Variable Optimization 65

Figure 2.2 Derivative undefined atx∗.

3.The theorem does not say what happens if a minimum or maximum occurs at

an endpoint of the interval of definition of the function. In this case

lim

h→ 0

f (x∗+ h)−f (x∗)

h

exists for positive values ofhonly or for negative values ofhonly, and hence

the derivative is not defined at the endpoints.

4.The theorem does not say that the function necessarily will have a minimum

or maximum at every point where the derivative is zero. For example, the

derivativef′( x)= 0 atx=0 for the function shown in Fig. 2.3. However, this

point is neither a minimum nor a maximum. In general, a pointx∗at which

f′(x∗) = 0 is called astationary point.

If the functionf (x)possesses continuous derivatives of every order that come in
question, in the neighborhood ofx=x∗, the following theorem provides the sufficient
condition for the minimum or maximum value of the function.

Figure 2.3 Stationary (inflection) point.